Modulo orientations with bounded out-degrees

Abstract: Let $G$ be a graph, let $k$ be a positive integer, and let $p:V(G)\rightarrow Z_k$ be a mapping with $|E(G)| \stackrel{k}{\equiv}\sum_{v\in V(G)}p(v) $. In this paper, we show that if $G$ is $(3k-3)$-edge-connected, then it has an orientation such that for each vertex $v$, $|d^+_G(v)-d_G(v)/2| < k$; also if $G$ contains $2k-2$ edge-disjoint spanning trees, then it admits such an orientation but by imposing greater out-degree bounds.